In tomographic imaging, a finite set of imaging samples are obtained of the underlying multi-dimensional function of interest. However, because of various physical restrictions of the sampling system, these samples are often obtained on nonuniform grids, thereby preventing the direct use and meaningful interpretation of these data. For example, in medical tomographic imaging such as the two-dimensional (2D) fan-beam computed tomography (CT), single-photon emission computed tomography (SPECT), positron emission tomography (PET), spiral (or helical) CT, diffraction tomography (DT), and magnetic resonance imaging (MRI), the acquired data are often sampled on nonuniform grids in the sinogram space, thus preventing the direct use of existing methods that are computationally efficient and numerically stable for reconstruction of tomographic images. In these situations, one can always use various multi-dimensional interpolation (MDI) methods to convert the samples that lie on nonuniform grids into samples that lie on uniform grids so that they can be processed directly and be presented meaningfully.
A wide variety of MDI methods have previously been developed. The methods that are based upon the Whittaker-Shannon sampling (WST) theorem can potentially provide accurate interpolation results. Unfortunately, these methods generally possesses the shortcoming of great computational burden, which increases drastically as the number of interpolation dimensions increases ("the curse of the dimensionality"). Attempts have been made to alleviate the computational burden by developing efficient interpolation methods. However, these methods are all associated with certain approximations. Virtually all of the previously developed methods calculate the desired uniform samples directly from the measured nonuniform samples, which generally requires the use of computationally burdensome algorithms if accuracy is to be preserved.